Spoke shave would be my choice as well. You could fine tune it by drilling a hole in a piece of scrap slightly larger than the finished diameter, line it with sand paper and use it to smooth the rough edges. On a piece like this, though, I’d think the spoke shave marks would be consistent with the rustic nature of the piece.

I wouldn’t go for perfectly round handles myself. To me that looks machine made, very precise. I would go for sort of round or even irregular shapes that subtly look like wear. If there is an original of that wheelbarrow I’m pretty sure it was never perfectly round.